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In [[number theory]], a '''Carmichael number''' is a [[composite number|composite]] positive [[integer]] <math>n</math> which satisfies the [[Modular arithmetic|congruence]]
{{distinguish|Dewey Decimal Classification}}
:<math>b^{n-1}\equiv 1\pmod{n}</math>
{{Numeral systems}}
for all integers <math>b</math> which are [[relatively prime]] to <math>n</math> (see [[modular arithmetic]]). They are named for [[Robert Daniel Carmichael|Robert Carmichael]]. The Carmichael numbers are the [[Knödel number]]s ''K''<sub>1</sub>.
The '''duodecimal''' system (also known as '''[[base (exponentiation)|base]]-12''' or '''dozenal''') is a [[positional notation]] [[numeral system]] using [[12 (number)|twelve]] as its [[radix|base]]. In this system, the number [[10 (number)|ten]] may be written as "A", "T" or "X", and the number [[11 (number)|eleven]] as "B" or "E" (another common notation, introduced by Sir [[Isaac Pitman]], is to use a rotated "2" for ten and a reversed "3" for eleven). The number twelve (that is, the number written as "12" in the [[decimal|base ten]] numerical system) is instead written as "10" in duodecimal (meaning "1 [[dozen]] and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (i.e. the same number that in decimal is written as "14"). Similarly, in duodecimal "100" means "1 [[gross (unit)|gross]]", "1000" means "1 [[great gross]]", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").


==Overview==
The number twelve, a [[highly composite number]], is the smallest number with four non-trivial [[integer factorization|factor]]s (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the [[subitizing and counting|subitizing]] range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal is sometimes named the number system with the most optimal [[radix economy]].{{citation needed|date=May 2012}} Of its factors, 2 and 3 are [[prime number|prime]], which means the [[multiplicative inverse|reciprocals]] of all [[smooth number|3-smooth]] numbers (such as 2, 3, 4, 6, 8, 9...) have a [[terminating decimal|terminating]] representation in duodecimal. In particular, the five most elementary fractions (½, ⅓, ⅔, ¼ and ¾) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (since it is the [[least common multiple]] of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the [[decimal]], [[vigesimal]], [[binary numeral system|binary]], [[octal]] and [[hexadecimal]] systems, although the [[sexagesimal]] system (where the reciprocals of all [[regular number|5-smooth]] numbers terminate) does better in this respect (but at the cost of an unwieldy multiplication table).
[[Fermat's little theorem]] states that all [[prime numbers]] have the above property. In this sense, Carmichael numbers are similar to prime numbers; in fact, they are called [[Fermat pseudoprime]]s. Carmichael numbers are sometimes also called '''absolute Fermat pseudoprimes'''.


Carmichael numbers are important because they pass the [[Fermat primality test]] but are not actually prime. Since Carmichael numbers exist, this primality test cannot be relied upon to prove the primality of a number, although it can still be used to prove a number is composite.
== Origin ==


Still, as numbers become larger, Carmichael numbers become very rare. For example, there are 20,138,200 Carmichael numbers between 1 and 10<sup>21</sup> (approximately one in 50 billion numbers).<ref name="Pinch2007">Richard Pinch, [http://s369624816.websitehome.co.uk/rgep/p82.pdf "The Carmichael numbers up to 10<sup>21</sup>"], May 2007.</ref> This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the [[Solovay-Strassen primality test]].
:''In this section, numerals are based on decimal [[Numerical digit|places]]. For example, 10 means [[10 (number)|ten]], 12 means [[12 (number)|twelve]].''


===Korselt's criterion===
Languages using duodecimal number systems are uncommon. Languages in the [[Nigeria]]n Middle Belt such as [[Janji]], [[Gbiri-Niragu]] (Kahugu), the Nimbia dialect of [[Gwandara]];<ref>{{citation
An alternative and equivalent definition of Carmichael numbers is given by '''Korselt's criterion'''.
| title=Decimal vs. Duodecimal: An interaction between two systems of numeration
| last=Matsushita
| first=Shuji
| journal=2nd Meeting of the AFLANG, October 1998, Tokyo
| year=1998
| url=http://web.archive.org/web/20081005230737/http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html
| accessdate=2011-05-29
}}</ref> the [[Chepang_language]] of [[Nepal]]<ref>{{citation
| contribution=Les principes de construction du nombre dans les langues tibéto-birmanes
| first=Martine
| last=Mazaudon
| title=La Pluralité
| editor-first=Jacques
| editor-last=François
| year=2002
| pages=91–119
| publisher=Peeters
| place=Leuven
| isbn=90-429-1295-2
| url=http://halshs.archives-ouvertes.fr/docs/00/16/68/91/PDF/numerationTB_SLP.pdf
}}</ref> and the [[Mahl language]] of [[Minicoy Island]] in [[India]] are known to use duodecimal numerals.  In fiction, [[J. R. R. Tolkien]]'s [[Elvish languages]] use a hybrid decimal-duodecimal system, primarily decimal but with special names for multiples of six.


:'''Theorem''' ([[Alwin Korselt|A. Korselt]] 1899): A positive composite integer <math>n</math> is a Carmichael number if and only if <math>n</math> is [[square-free integer|square-free]], and for all [[prime divisor]]s <math>p</math> of <math>n</math>, it is true that <math>p - 1 \mid n - 1</math> (where <math>a \mid b</math> means that <math>a</math> [[divisor|divides]] <math>b</math>).
[[Germanic languages]] have special words for 11 and 12, such as ''eleven'' and ''twelve'' in [[English language|English]], which are often misinterpreted as vestiges of a duodecimal system.{{citation needed|date=January 2012}}  However, they are considered to come from [[Proto-Germanic]] *''ainlif'' and *''twalif'' (respectively ''one left'' and ''two left''), both of which were decimal.


It follows from this theorem that all Carmichael numbers are [[odd number|odd]], since any even composite number that is square-free (and hence has only one prime factor of two) will have at least one odd prime factor, and thus <math>p - 1  \mid n - 1</math> results in an even dividing an odd, a contradiction. (The oddness of Carmichael numbers also follows from the fact that <math>-1</math> is a [[Fermat primality test|Fermat witness]] for any even number.)
Historically, [[Unit of measurement|unit]]s of [[time]] in many [[civilization]]s are duodecimal. There are twelve signs of the [[zodiac]], twelve months in a year, and the [[Babylonians]] had twelve hours in a day (although at some point this was changed to 24). Traditional [[Chinese calendar]]s, clocks, and compasses are based on the twelve [[Earthly Branches]].
From the criterion it also follows that Carmichael numbers are [[Cyclic number (group theory)|cyclic]].<ref>[http://www.numericana.com/data/crump.htm Carmichael Multiples of Odd Cyclic Numbers] "Any divisor of a Carmichael number must be an odd cyclic number"</ref><ref>Proof sketch: If <math>n</math> is square-free but not cyclic, <math>p_i \mid p_j - 1</math> for two prime factors <math>p_i</math> and <math>p_j</math> of <math>n</math>. But if <math>n</math> satisfies Korselt then <math>p_j - 1  \mid n - 1</math>, so by transitivity of the "divides" relation <math>p_i  \mid n - 1</math>. But <math>p_i</math> is also a factor of <math>n</math>, a contradiction.</ref>


==Discovery==
Being a versatile denominator in fractions may explain why we have 12 inches in an imperial foot, 12 ounces in a [[troy weight|troy]] pound, 12 [[British One Penny coin (pre-decimal)|old British pence]] in a [[shilling]], 24 (12×2) hours in a day, and many other items counted by the [[dozen]], [[gross (unit)|gross]] ([[144 (number)|144]], [[square number|square]] of 12) or [[great gross]] ([[1728 (number)|1728]], [[cube (arithmetic)|cube]] of 12). The Romans used a fraction system based on 12, including the [[uncia (length)|uncia]] which became both the English words ''[[ounce]]'' and ''inch''. Pre-[[Decimal Day|decimalisation]], the [[United Kingdom]] and [[Republic of Ireland]] used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the [[pound sterling]] or [[Irish pound]]), and [[Charlemagne]] established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.
Korselt was the first who observed the basic properties of Carmichael numbers, but he could not find any examples. In 1910, Carmichael<ref name="Carmichael1910">{{cite journal |author=R. D. Carmichael|title=Note on a new number theory function |journal=Bulletin of the American Mathematical Society |volume=16 |issue=5|year=1910 |pages=232–238 |url=http://www.ams.org/journals/bull/1910-16-05/home.html}}</ref> found the first and smallest such number, 561, which explains the name "Carmichael number".


That 561 is a Carmichael number can be seen with Korselt's criterion. Indeed, <math>561 = 3 \cdot 11 \cdot 17</math> is square-free and <math>2 | 560</math>, <math>10 | 560</math> and <math>16 | 560</math>.
The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones ([[Phalanx bone|phalanges]]) on one hand (three on each of four fingers).<ref>{{citation
| title=ヒマラヤの満月と十二進法 (The Full Moon in the Himalayas and the Duodecimal System)
| last=Nishikawa
| first=Yoshiaki
| year=2002
| url=http://www.kankyok.co.jp/nue/nue11/nue11_01.html
| accessdate=2008-03-24
}}</ref> It is possible to count to 12 with your thumb acting as a pointer, touching each finger bone in turn. A traditional [[finger counting]] system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.<ref name=Ifrah>{{Citation
  | last = Ifrah
  | first = Georges
  | author-link = Georges Ifrah
  | title = The Universal History of Numbers: From prehistory to the invention of the computer.
  | publisher = [[John Wiley and Sons]]
  | year= 2000
  | page =
  | isbn = 0-471-39340-1
}}. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.</ref><ref name=Macey>{{Citation|last=Macey|first=Samuel L.|title=The Dynamics of Progress: Time, Method, and Measure|year=1989|publisher=University of Georgia Press|location=Atlanta, Georgia|isbn=978-0-8203-3796-8|pages=92|url=http://books.google.com/books?id=xlzCWmXguwsC&pg=PA92&lpg=PA92}}</ref>


The next six Carmichael numbers are {{OEIS|id=A002997}}:
== Places ==
:<math>1105 = 5 \cdot 13 \cdot 17 \qquad (4 \mid 1104; 12 \mid 1104; 16 \mid 1104)</math>
In a duodecimal place system, [[10 (number)|ten]] can be written as A, [[11 (number)|eleven]] can be written as B, and twelve is written as 10. For alternative symbols, see [[#Advocacy and "dozenalism"|below]].
:<math>1729 = 7 \cdot 13 \cdot 19 \qquad (6 \mid 1728; 12 \mid 1728; 18 \mid 1728)</math>
:<math>2465 = 5 \cdot 17 \cdot 29 \qquad (4 \mid 2464; 16 \mid 2464; 28 \mid 2464)</math>
:<math>2821 = 7 \cdot 13 \cdot 31 \qquad (6 \mid 2820; 12 \mid 2820; 30 \mid 2820)</math>
:<math>6601 = 7 \cdot 23 \cdot 41 \qquad (6 \mid 6600; 22 \mid 6600; 40 \mid 6600)</math>
:<math>8911 = 7 \cdot 19 \cdot 67 \qquad (6 \mid 8910; 18 \mid 8910; 66 \mid 8910).</math>


These first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician [[Václav Šimerka]] in 1885<ref name="Simerka1885">{{cite journal |author=V. Šimerka|title=Zbytky z arithmetické posloupnosti (On the remainders of an arithmetic progression) |journal=Časopis pro pěstování matematiky a fysiky |volume=14 |issue=5|year=1885 |pages=221–225 |url=http://dml.cz/handle/10338.dmlcz/122245}}</ref> (thus preceding not just Carmichael but also Korselt, although Šimerka did not find anything like Korselt's criterion). His work, however, remained unnoticed.
According to this notation, duodecimal 50 expresses the same quantity as decimal [[60 (number)|60]] (= five times twelve), duodecimal 60 is equivalent to decimal [[72 (number)|72]] (= six times twelve = half a gross), duodecimal 100 has the same value as decimal [[144 (number)|144]] (= twelve times twelve = one gross), etc.


J. Chernick<ref name="Chernick1939">{{cite journal |author=Chernick, J. |title=On Fermat's simple theorem |journal=Bull. Amer. Math. Soc. |volume=45 |year=1939 |pages=269–274 |doi=10.1090/S0002-9904-1939-06953-X  |url=http://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf}}</ref> proved a theorem in 1939 which can be used to construct a [[subset]] of Carmichael numbers. The number <math>(6k + 1)(12k + 1)(18k + 1)</math> is a Carmichael number if its three factors are all prime. Whether this formula produces an infinite quantity of Carmichael numbers is an open question (though it is implied by [[Dickson's conjecture]]).
== Comparison to other numeral systems ==


[[Paul Erdős]] heuristically argued there should be infinitely many Carmichael numbers. In 1994 it was shown by [[W. R. (Red) Alford]], [[Andrew Granville]] and [[Carl Pomerance]] that there really do exist infinitely many Carmichael numbers. Specifically, they showed that for sufficiently large <math>n</math>, there are at least <math>n^{2/7}</math> Carmichael numbers between 1 and <math>n</math>.<ref name="Alford1994">{{cite journal |author=W. R. Alford, A. Granville, C. Pomerance |title=There are Infinitely Many Carmichael Numbers |journal=[[Annals of Mathematics]] |volume=139 |year=1994 |pages=703–722 |doi=10.2307/2118576 |url=http://www.math.dartmouth.edu/~carlp/PDF/paper95.pdf}}</ref>
{| class="wikitable" style="float:right; text-align:center"
|+ A duodecimal [[multiplication table]]
|-
! style="width:1.4em" | 1
! style="width:1.4em" | 2
! style="width:1.4em" | 3
! style="width:1.4em" | 4
! style="width:1.4em" | 5
! style="width:1.4em" | 6
! style="width:1.4em" | 7
! style="width:1.4em" | 8
! style="width:1.4em" | 9
! style="width:1.4em" | A
! style="width:1.4em" | B
! style="width:1.4em" | 10
|-
! 2
| 4 || 6|| 8 || A || 10 || 12 || 14 || 16 || 18 || 1A || 20
|-
! 3
| 6 || 9 || 10 || 13 || 16 || 19 || 20 || 23 || 26 || 29 || 30
|-
! 4
| 8 || 10 || 14 || 18 || 20 || 24 || 28 || 30 || 34 || 38 || 40
|-
! 5
| A || 13 || 18 || 21 || 26 || 2B || 34 || 39 || 42 || 47 || 50
|-
! 6
| 10 || 16 || 20 || 26 || 30 || 36 || 40 || 46 || 50 || 56 || 60
|-
! 7
| 12 || 19 || 24 || 2B || 36 || 41 || 48 || 53 || 5A || 65 || 70
|-
! 8
| 14 || 20 || 28 || 34 || 40 || 48 || 54 || 60 || 68 || 74 || 80
|-
! 9
| 16 || 23 || 30 || 39 || 46 || 53 || 60 || 69 || 76 || 83 || 90
|-
! A
| 18 || 26 || 34 || 42 || 50 || 5A || 68 || 76 || 84 || 92 || A0
|-
! B
| 1A || 29 || 38 || 47 || 56 || 65 || 74 || 83 || 92 || A1 || B0
|-
! 10
| 20 || 30 || 40 || 50 || 60 || 70 || 80 || 90 || A0 || B0 || 100
|}
The number 12 has six factors, which are [[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[6 (number)|6]], and [[12 (number)|12]], of which 2 and 3 are [[prime number|prime]]. The decimal system has only four factors, which are [[1 (number)|1]], [[2 (number)|2]], [[5 (number)|5]], and [[10 (number)|10]]; of which 2 and 5 are prime. Vigesimal adds two factors to those of ten, namely [[4 (number)|4]] and [[20 (number)|20]], but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base (i.e., the digit set and the multiplication table are much larger). Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal has five factors, adding 4, [[8 (number)|8]] and [[16 (number)|16]] to those of 2, but no additional prime. [[Base 30|Trigesimal]] is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30), the smallest system that has four different prime factors is Base 210 and the pattern follows the [[primorial]]s. [[Sexagesimal]]—which the ancient [[Sumerians]] and [[Babylonia]]ns among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors.{{-}}<!-- the {{-}} template keeps the multiplication table from squeezing the heading for the next section-->
 
== Conversion tables to and from decimal ==
 
To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under [[Positional notation#Base conversion|positional notation]]). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any dozenal number between 0.01 and BBB,BBB.BB to decimal, or any decimal number between 0.01 and 999,999.99 to dozenal. To use them, we first decompose the given number into a sum of numbers with only one significant digit each. For example:
 
123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08
 
This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then we use the digit conversion tables to obtain the equivalent value in the target base for each digit. If the given number is in dozenal and the target base is decimal, we get:
 
<small>(dozenal)</small> 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = <small>(decimal)</small> 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58{{overline|3}}333333333... + 0.0{{overline|5}}5555555555...
 
Now, since the summands are already converted to base ten, we use the usual decimal arithmetic to perform the addition and recompose the number, arriving at the conversion result:
 
  Dozenal  ----->  Decimal
 
  100,000    =    248,832
    20,000    =    41,472
    3,000    =      5,184
      400    =        576
        50    =        60
  +      6    =  +      6
        0.7  =          0.58{{overline|3}}333333333...
        0.08  =          0.0{{overline|5}}5555555555...
--------------------------------------------
  123,456.78  =    296,130.63{{overline|8}}888888888...
 
That is, <small>(dozenal)</small> 123,456.78 equals <small>(decimal)</small> 296,130.63{{overline|8}}888888888... ≈ 296,130.64
 
If the given number is in decimal and the target base is dozenal, the method is basically same. Using the digit conversion tables:
 
<small>(decimal)</small> 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = <small>(dozenal)</small> 49,A54 + B,6A8 + 1,8A0 + 294 + 42 + 6 + 0.8{{overline|4972}}4972497249724972497... + 0.{{overline|0B62A68781B05915343A}}0B62...
 
However, in order to do this sum and recompose the number, we now have to use the addition tables for dozenal, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in dozenal as well. In decimal, 6 + 6 equals 12, but in dozenal it equals 10; so if we used decimal arithmetic with dozenal numbers we would arrive at an incorrect result. Doing the arithmetic properly in dozenal, we get the result:
 
  Decimal  ----->  Dozenal
 
  100,000    =    49,A54
    20,000    =      B,6A8
    3,000    =     1,8A0
      400    =       294
        50    =         42
  +      6    =   +      6
        0.7  =         0.8{{overline|4972}}4972497249724972497...
        0.08  =         0.{{overline|0B62A68781B05915343A}}0B62...
--------------------------------------------------------
  123,456.78  =    5B,540.9{{overline|43A0B62A68781B059153}}43A...


Löh and Niebuhr in 1992 found some huge Carmichael numbers, including one with 1,101,518 factors and over 16 million digits.
That is, <small>(decimal)</small> 123,456.78 equals <small>(dozenal)</small> 5B,540.9{{overline|43A0B62A68781B059153}}43A... ≈ 5B,540.94


==Properties==
=== Dozenal to decimal digit conversion ===
=== Factorizations ===
Carmichael numbers have at least three positive prime factors. The first Carmichael numbers with <math>k = 3, 4, 5, \ldots</math> prime factors are {{OEIS|id=A006931}}:


{| class="wikitable"
{|class="wikitable"
|-
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
|-
| bgcolor=#c0c0c0 | '''100,000'''
| 248,832
| bgcolor=#c0c0c0 | '''10,000'''
| 20,736
| bgcolor=#c0c0c0 | '''1,000'''
| 1,728
| bgcolor=#c0c0c0 | '''100'''
| 144
| bgcolor=#c0c0c0 | '''10'''
| 12
| bgcolor=#c0c0c0 | '''1'''
| 1
| bgcolor=#c0c0c0 | '''0.1'''
| 0.08{{overline|3}}
| bgcolor=#c0c0c0 | '''0.01'''
| 0.0069{{overline|4}}
|-
| bgcolor=#c0c0c0 | '''200,000'''
| 497,664
| bgcolor=#c0c0c0 | '''20,000'''
| 41,472
| bgcolor=#c0c0c0 | '''2,000'''
| 3,456
| bgcolor=#c0c0c0 | '''200'''
| 288
| bgcolor=#c0c0c0 | '''20'''
| 24
| bgcolor=#c0c0c0 | '''2'''
| 2
| bgcolor=#c0c0c0 | '''0.2'''
| 0.1{{overline|6}}
| bgcolor=#c0c0c0 | '''0.02'''
| 0.013{{overline|8}}
|-
|-
!''k'' !!&nbsp;
| bgcolor=#c0c0c0 | '''300,000'''
| 746,496
| bgcolor=#c0c0c0 | '''30,000'''
| 62,208
| bgcolor=#c0c0c0 | '''3,000'''
| 5,184
| bgcolor=#c0c0c0 | '''300'''
| 432
| bgcolor=#c0c0c0 | '''30'''
| 36
| bgcolor=#c0c0c0 | '''3'''
| 3
| bgcolor=#c0c0c0 | '''0.3'''
| 0.25
| bgcolor=#c0c0c0 | '''0.03'''
| 0.0208{{overline|3}}
|-
|-
| 3 || <math>561 = 3 \cdot 11 \cdot 17\,</math>
| bgcolor=#c0c0c0 | '''400,000'''
| 995,328
| bgcolor=#c0c0c0 | '''40,000'''
| 82,944
| bgcolor=#c0c0c0 | '''4,000'''
| 6,912
| bgcolor=#c0c0c0 | '''400'''
| 576
| bgcolor=#c0c0c0 | '''40'''
| 48
| bgcolor=#c0c0c0 | '''4'''
| 4
| bgcolor=#c0c0c0 | '''0.4'''
| 0.{{overline|3}}
| bgcolor=#c0c0c0 | '''0.04'''
| 0.02{{overline|7}}
|-
|-
| 4 || <math>41041 = 7 \cdot 11 \cdot 13 \cdot 41\,</math>
| bgcolor=#c0c0c0 | '''500,000'''
| 1,244,160
| bgcolor=#c0c0c0 | '''50,000'''
| 103,680
| bgcolor=#c0c0c0 | '''5,000'''
| 8,640
| bgcolor=#c0c0c0 | '''500'''
| 720
| bgcolor=#c0c0c0 | '''50'''
| 60
| bgcolor=#c0c0c0 | '''5'''
| 5
| bgcolor=#c0c0c0 | '''0.5'''
| 0.41{{overline|6}}
| bgcolor=#c0c0c0 | '''0.05'''
| 0.0347{{overline|2}}
|-
| bgcolor=#c0c0c0 | '''600,000'''
| 1,492,992
| bgcolor=#c0c0c0 | '''60,000'''
| 124,416
| bgcolor=#c0c0c0 | '''6,000'''
| 10,368
| bgcolor=#c0c0c0 | '''600'''
| 864
| bgcolor=#c0c0c0 | '''60'''
| 72
| bgcolor=#c0c0c0 | '''6'''
| 6
| bgcolor=#c0c0c0 | '''0.6'''
| 0.5
| bgcolor=#c0c0c0 | '''0.06'''
| 0.041{{overline|6}}
|-
|-
| 5 || <math>825265 = 5 \cdot 7 \cdot 17 \cdot 19 \cdot 73\,</math>
| bgcolor=#c0c0c0 | '''700,000'''
| 1,741,824
| bgcolor=#c0c0c0 | '''70,000'''
| 145,152
| bgcolor=#c0c0c0 | '''7,000'''
| 12,096
| bgcolor=#c0c0c0 | '''700'''
| 1008
| bgcolor=#c0c0c0 | '''70'''
| 84
| bgcolor=#c0c0c0 | '''7'''
| 7
| bgcolor=#c0c0c0 | '''0.7'''
| 0.58{{overline|3}}
| bgcolor=#c0c0c0 | '''0.07'''
| 0.0486{{overline|1}}
|-
|-
| 6 || <math>321197185 = 5 \cdot 19 \cdot 23 \cdot 29 \cdot 37 \cdot 137\,</math>
| bgcolor=#c0c0c0 | '''800,000'''
| 1,990,656
| bgcolor=#c0c0c0 | '''80,000'''
| 165,888
| bgcolor=#c0c0c0 | '''8,000'''
| 13,824
| bgcolor=#c0c0c0 | '''800'''
| 1152
| bgcolor=#c0c0c0 | '''80'''
| 96
| bgcolor=#c0c0c0 | '''8'''
| 8
| bgcolor=#c0c0c0 | '''0.8'''
| 0.{{overline|6}}
| bgcolor=#c0c0c0 | '''0.08'''
| 0.0{{overline|5}}
|-
|-
| 7 || <math>5394826801 = 7 \cdot 13 \cdot 17 \cdot 23 \cdot 31 \cdot 67 \cdot 73\,</math>
| bgcolor=#c0c0c0 | '''900,000'''
| 2,239,488
| bgcolor=#c0c0c0 | '''90,000'''
| 186,624
| bgcolor=#c0c0c0 | '''9,000'''
| 15,552
| bgcolor=#c0c0c0 | '''900'''
| 1,296
| bgcolor=#c0c0c0 | '''90'''
| 108
| bgcolor=#c0c0c0 | '''9'''
| 9
| bgcolor=#c0c0c0 | '''0.9'''
| 0.75
| bgcolor=#c0c0c0 | '''0.09'''
| 0.0625
|-
|-
| 8 || <math>232250619601 = 7 \cdot 11 \cdot 13 \cdot 17 \cdot 31 \cdot 37 \cdot 73 \cdot 163\,</math>
| bgcolor=#c0c0c0 | '''A00,000'''
| 2,488,320
| bgcolor=#c0c0c0 | '''A0,000'''
| 207,360
| bgcolor=#c0c0c0 | '''A,000'''
| 17,280
| bgcolor=#c0c0c0 | '''A00'''
| 1,440
| bgcolor=#c0c0c0 | '''A0'''
| 120
| bgcolor=#c0c0c0 | '''A'''
| 10
| bgcolor=#c0c0c0 | '''0.A'''
| 0.8{{overline|3}}
| bgcolor=#c0c0c0 | '''0.0A'''
| 0.069{{overline|4}}
|-
|-
| 9 || <math>9746347772161 = 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 31 \cdot 37 \cdot 41 \cdot 641\,</math>
| bgcolor=#c0c0c0 | '''B00,000'''
| 2,737,152
| bgcolor=#c0c0c0 | '''B0,000'''
| 228,096
| bgcolor=#c0c0c0 | '''B,000'''
| 19,008
| bgcolor=#c0c0c0 | '''B00'''
| 1,584
| bgcolor=#c0c0c0 | '''B0'''
| 132
| bgcolor=#c0c0c0 | '''B'''
| 11
| bgcolor=#c0c0c0 | '''0.B'''
| 0.91{{overline|6}}
| bgcolor=#c0c0c0 | '''0.0B'''
| 0.0763{{overline|8}}
|}
|}


The first Carmichael numbers with 4 prime factors are {{OEIS|id=A074379}}:
=== Decimal to dozenal digit conversion ===


{| class="wikitable"
{|class="wikitable"
|-
|-
!''i'' !!&nbsp;
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
|-
|-
| 1 || <math>41041 = 7 \cdot 11 \cdot 13 \cdot 41\,</math>
| bgcolor=#c0c0c0 | '''100,000'''
| 49,A54
| bgcolor=#c0c0c0 | '''10,000'''
| 5,954
| bgcolor=#c0c0c0 | '''1,000'''
| 6B4
| bgcolor=#c0c0c0 | '''100'''
| 84
| bgcolor=#c0c0c0 | '''10'''
| A
| bgcolor=#c0c0c0 | '''1'''
| 1
| bgcolor=#c0c0c0 | '''0.1'''
| 0.1{{overline|2497}}
| bgcolor=#c0c0c0 | '''0.01'''
| 0.0{{overline|15343A0B62A68781B059}}
|-
|-
| 2 || <math>62745 = 3 \cdot 5 \cdot 47 \cdot 89\,</math>
| bgcolor=#c0c0c0 | '''200,000'''
| 97,8A8
| bgcolor=#c0c0c0 | '''20,000'''
| B,6A8
| bgcolor=#c0c0c0 | '''2,000'''
| 1,1A8
| bgcolor=#c0c0c0 | '''200'''
| 148
| bgcolor=#c0c0c0 | '''20'''
| 18
| bgcolor=#c0c0c0 | '''2'''
| 2
| bgcolor=#c0c0c0 | '''0.2'''
| 0.{{overline|2497}}
| bgcolor=#c0c0c0 | '''0.02'''
| 0.0{{overline|2A68781B05915343A0B6}}
|-
|-
| 3 || <math>63973 = 7 \cdot 13 \cdot 19 \cdot 37\,</math>
| bgcolor=#c0c0c0 | '''300,000'''
| 125,740
| bgcolor=#c0c0c0 | '''30,000'''
| 15,440
| bgcolor=#c0c0c0 | '''3,000'''
| 1,8A0
| bgcolor=#c0c0c0 | '''300'''
| 210
| bgcolor=#c0c0c0 | '''30'''
| 26
| bgcolor=#c0c0c0 | '''3'''
| 3
| bgcolor=#c0c0c0 | '''0.3'''
| 0.3{{overline|7249}}
| bgcolor=#c0c0c0 | '''0.03'''
| 0.0{{overline|43A0B62A68781B059153}}
|-
|-
| 4 || <math>75361 = 11 \cdot 13 \cdot 17 \cdot 31\,</math>
| bgcolor=#c0c0c0 | '''400,000'''
| 173,594
| bgcolor=#c0c0c0 | '''40,000'''
| 1B,194
| bgcolor=#c0c0c0 | '''4,000'''
| 2,394
| bgcolor=#c0c0c0 | '''400'''
| 294
| bgcolor=#c0c0c0 | '''40'''
| 34
| bgcolor=#c0c0c0 | '''4'''
| 4
| bgcolor=#c0c0c0 | '''0.4'''
| 0.{{overline|4972}}
| bgcolor=#c0c0c0 | '''0.04'''
| 0.0{{overline|5915343A0B62A68781B0}}
|-
|-
| 5 || <math>101101 = 7 \cdot 11 \cdot 13 \cdot 101\,</math>
| bgcolor=#c0c0c0 | '''500,000'''
| 201,428
| bgcolor=#c0c0c0 | '''50,000'''
| 24,B28
| bgcolor=#c0c0c0 | '''5,000'''
| 2,A88
| bgcolor=#c0c0c0 | '''500'''
| 358
| bgcolor=#c0c0c0 | '''50'''
| 42
| bgcolor=#c0c0c0 | '''5'''
| 5
| bgcolor=#c0c0c0 | '''0.5'''
| 0.6
| bgcolor=#c0c0c0 | '''0.05'''
| 0.0{{overline|7249}}
|-
|-
| 6 || <math>126217 = 7 \cdot 13 \cdot 19 \cdot 73\,</math>
| bgcolor=#c0c0c0 | '''600,000'''
| 24B,280
| bgcolor=#c0c0c0 | '''60,000'''
| 2A,880
| bgcolor=#c0c0c0 | '''6,000'''
| 3,580
| bgcolor=#c0c0c0 | '''600'''
| 420
| bgcolor=#c0c0c0 | '''60'''
| 50
| bgcolor=#c0c0c0 | '''6'''
| 6
| bgcolor=#c0c0c0 | '''0.6'''
| 0.{{overline|7249}}
| bgcolor=#c0c0c0 | '''0.06'''
| 0.0{{overline|8781B05915343A0B62A6}}
|-
|-
| 7 || <math>172081 = 7 \cdot 13 \cdot 31 \cdot 61\,</math>
| bgcolor=#c0c0c0 | '''700,000'''
| 299,114
| bgcolor=#c0c0c0 | '''70,000'''
| 34,614
| bgcolor=#c0c0c0 | '''7,000'''
| 4,074
| bgcolor=#c0c0c0 | '''700'''
| 4A4
| bgcolor=#c0c0c0 | '''70'''
| 5A
| bgcolor=#c0c0c0 | '''7'''
| 7
| bgcolor=#c0c0c0 | '''0.7'''
| 0.8{{overline|4972}}
| bgcolor=#c0c0c0 | '''0.07'''
| 0.0{{overline|A0B62A68781B05915343}}
|-
|-
| 8 || <math>188461 = 7 \cdot 13 \cdot 19 \cdot 109\,</math>
| bgcolor=#c0c0c0 | '''800,000'''
| 326,B68
| bgcolor=#c0c0c0 | '''80,000'''
| 3A,368
| bgcolor=#c0c0c0 | '''8,000'''
| 4,768
| bgcolor=#c0c0c0 | '''800'''
| 568
| bgcolor=#c0c0c0 | '''80'''
| 68
| bgcolor=#c0c0c0 | '''8'''
| 8
| bgcolor=#c0c0c0 | '''0.8'''
| 0.{{overline|9724}}
| bgcolor=#c0c0c0 | '''0.08'''
| 0.{{overline|0B62A68781B05915343A}}
|-
|-
| 9 || <math>278545 = 5 \cdot 17 \cdot 29 \cdot 113\,</math>
| bgcolor=#c0c0c0 | '''900,000'''
|-
| 374,A00
| 10 || <math>340561 = 13 \cdot 17 \cdot 23 \cdot 67\,</math>
| bgcolor=#c0c0c0 | '''90,000'''
| 44,100
| bgcolor=#c0c0c0 | '''9,000'''
| 5,260
| bgcolor=#c0c0c0 | '''900'''
| 630
| bgcolor=#c0c0c0 | '''90'''
| 76
| bgcolor=#c0c0c0 | '''9'''
| 9
| bgcolor=#c0c0c0 | '''0.9'''
| 0.A{{overline|9724}}
| bgcolor=#c0c0c0 | '''0.09'''
| 0.1{{overline|0B62A68781B05915343A}}
|}
|}


The second Carmichael number (1105) can be expressed as the sum of two squares in more ways than any smaller number. The third Carmichael number ([[1729 (number)|1729]]) is the Hardy-Ramanujan Number: the smallest number that can be expressed as the sum of two cubes in two different ways.
=== Conversion of powers ===


===Distribution===
{|class="wikitable"
 
|-
Let <math>C(X)</math> denote the number of Carmichael numbers less than or equal to <math>X</math>. The distribution of Carmichael numbers by powers of 10:<ref name="Pinch2007"/>
| rowspan="2" | ''Exponent''
 
| colspan="2" | Powers of 2
<center>
| colspan="2" | Powers of 3
{| class="wikitable"
| colspan="2" | Powers of 4
| colspan="2" | Powers of 5
| colspan="2" | Powers of 6
| colspan="2" | Powers of 7
|-
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
|-
| ''^6''
| bgcolor=#c0c0c0 | '''64'''
| 54
| bgcolor=#c0c0c0 | '''729'''
| 509
| bgcolor=#c0c0c0 | '''4,096'''
| 2454
| bgcolor=#c0c0c0 | '''15,625'''
| 9,061
| bgcolor=#c0c0c0 | '''46,656'''
| 23,000
| bgcolor=#c0c0c0 | '''117,649'''
| 58,101
|-
| ''^5''
| bgcolor=#c0c0c0 | '''32'''
| 28
| bgcolor=#c0c0c0 | '''243'''
| 183
| bgcolor=#c0c0c0 | '''1,024'''
| 714
| bgcolor=#c0c0c0 | '''3,125'''
| 1,985
| bgcolor=#c0c0c0 | '''7,776'''
| 4,600
| bgcolor=#c0c0c0 | '''16,807'''
| 9,887
|-
| ''^4''
| bgcolor=#c0c0c0 | '''16'''
| 14
| bgcolor=#c0c0c0 | '''81'''
| 69
| bgcolor=#c0c0c0 | '''256'''
| 194
| bgcolor=#c0c0c0 | '''625'''
| 441
| bgcolor=#c0c0c0 | '''1,296'''
| 900
| bgcolor=#c0c0c0 | '''2,401'''
| 1,481
|-
| ''^3''
| bgcolor=#c0c0c0 | '''8'''
| 8
| bgcolor=#c0c0c0 | '''27'''
| 23
| bgcolor=#c0c0c0 | '''64'''
| 54
| bgcolor=#c0c0c0 | '''125'''
| A5
| bgcolor=#c0c0c0 | '''216'''
| 160
| bgcolor=#c0c0c0 | '''343'''
| 247
|-
| ''^2''
| bgcolor=#c0c0c0 | '''4'''
| 4
| bgcolor=#c0c0c0 | '''9'''
| 9
| bgcolor=#c0c0c0 | '''16'''
| 14
| bgcolor=#c0c0c0 | '''25'''
| 21
| bgcolor=#c0c0c0 | '''36'''
| 30
| bgcolor=#c0c0c0 | '''49'''
| 41
|-
|-
! <math>n</math>
| ''^1''
| bgcolor=#c0c0c0 | '''2'''
| 2
| bgcolor=#c0c0c0 | '''3'''
| 3
| 3
| bgcolor=#c0c0c0 | '''4'''
| 4
| 4
| bgcolor=#c0c0c0 | '''5'''
| 5
| 5
| bgcolor=#c0c0c0 | '''6'''
| 6
| 6
| bgcolor=#c0c0c0 | '''7'''
| 7
| 7
|-
| ''^−1''
| bgcolor=#c0c0c0 | '''0.5'''
| 0.6
| bgcolor=#c0c0c0 | '''0.{{overline|3}}'''
| 0.4
| bgcolor=#c0c0c0 | '''0.25'''
| 0.3
| bgcolor=#c0c0c0 | '''0.2'''
| 0.{{overline|2497}}
| bgcolor=#c0c0c0 | '''0.1{{overline|6}}'''
| 0.2
| bgcolor=#c0c0c0 | '''0.{{overline|142857}}'''
| 0.{{overline|186A35}}
|-
| ''^−2''
| bgcolor=#c0c0c0 | '''0.25'''
| 0.3
| bgcolor=#c0c0c0 | '''0.{{overline|1}}'''
| 0.14
| bgcolor=#c0c0c0 | '''0.0625'''
| 0.09
| bgcolor=#c0c0c0 | '''0.04'''
| 0.{{overline|05915343A0<br/>B62A68781B}}
| bgcolor=#c0c0c0 | '''0.02{{overline|7}}'''
| 0.04
| bgcolor=#c0c0c0 | '''0.{{overline|0204081632653<br/>06122448979591<br/>836734693877551}}'''
| 0.{{overline|02B322547A05A<br/>644A9380B908996<br/>741B615771283B}}
|}
{|class="wikitable"
|-
| rowspan="2" | ''Exponent''
| colspan="2" | Powers of 8
| colspan="2" | Powers of 9
| colspan="2" | '''Powers of 10'''
| colspan="2" | Powers of 11
| colspan="2" | '''Powers of 12'''
|-
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
|-
| ''^6''
| bgcolor=#c0c0c0 | '''262,144'''
| 107,854
| bgcolor=#c0c0c0 | '''531,441'''
| 217,669
| bgcolor=#c0c0c0 | '''1,000,000'''
| 402,854
| bgcolor=#c0c0c0 | '''1,771,561'''
| 715,261
| bgcolor=#c0c0c0 | '''2,985,984'''
| 1,000,000
|-
| ''^5''
| bgcolor=#c0c0c0 | '''32,768'''
| 16,B68
| bgcolor=#c0c0c0 | '''59,049'''
| 2A,209
| bgcolor=#c0c0c0 | '''100,000'''
| 49,A54
| bgcolor=#c0c0c0 | '''161,051'''
| 79,24B
| bgcolor=#c0c0c0 | '''248,832'''
| 100,000
|-
| ''^4''
| bgcolor=#c0c0c0 | '''4,096'''
| 2,454
| bgcolor=#c0c0c0 | '''6,561'''
| 3,969
| bgcolor=#c0c0c0 | '''10,000'''
| 5,954
| bgcolor=#c0c0c0 | '''14,641'''
| 8,581
| bgcolor=#c0c0c0 | '''20,736'''
| 10,000
|-
| ''^3''
| bgcolor=#c0c0c0 | '''512'''
| 368
| bgcolor=#c0c0c0 | '''729'''
| 509
| bgcolor=#c0c0c0 | '''1,000'''
| 6B4
| bgcolor=#c0c0c0 | '''1,331'''
| 92B
| bgcolor=#c0c0c0 | '''1,728'''
| 1,000
|-
| ''^2''
| bgcolor=#c0c0c0 | '''64'''
| 54
| bgcolor=#c0c0c0 | '''81'''
| 69
| bgcolor=#c0c0c0 | '''100'''
| 84
| bgcolor=#c0c0c0 | '''121'''
| A1
| bgcolor=#c0c0c0 | '''144'''
| 100
|-
| ''^1''
| bgcolor=#c0c0c0 | '''8'''
| 8
| 8
| bgcolor=#c0c0c0 | '''9'''
| 9
| 9
| bgcolor=#c0c0c0 | '''10'''
| A
| bgcolor=#c0c0c0 | '''11'''
| B
| bgcolor=#c0c0c0 | '''12'''
| 10
| 10
| 11
| 12
| 13
| 14
| 15
| 16
| 17
| 18
| 19
| 20
| 21
|-
|-
! <math>C(10^n)</math>
| ''^−1''
| 1
| bgcolor=#c0c0c0 | '''0.125'''
| 7
| 0.16
| 16
| bgcolor=#c0c0c0 | '''0.{{overline|1}}'''
| 43
| 0.14
| 105
| bgcolor=#c0c0c0 | '''0.1'''
| 255
| 0.1{{overline|2497}}
| 646
| bgcolor=#c0c0c0 | '''0.{{overline|09}}'''
| 1547
| 0.{{overline|1}}
| 3605
| bgcolor=#c0c0c0 | '''0.08{{overline|3}}'''
| 8241
| 0.1
| 19279
|-
| 44706
| ''^−2''
| 105212
| bgcolor=#c0c0c0 | '''0.015625'''
| 246683
| 0.023
| 585355
| bgcolor=#c0c0c0 | '''0.{{overline|012345679}}'''
| 1401644
| 0.0194
| 3381806
| bgcolor=#c0c0c0 | '''0.01'''
| 8220777
| 0.0{{overline|15343A0B6<br/>2A68781B059}}
| 20138200
| bgcolor=#c0c0c0 | '''0.{{overline|00826446280<br/>99173553719}}'''
| 0.{{overline|0123456789B}}
| bgcolor=#c0c0c0 | '''0.0069{{overline|4}}'''
| 0.01
|}
 
==Fractions and irrational numbers==
===Fractions===
Duodecimal [[Fraction (mathematics)|fraction]]s may be simple:
 
* {{frac|2}} = 0.6
* {{frac|3}} = 0.4
* {{frac|4}} = 0.3
* {{frac|6}} = 0.2
* {{frac|8}} = 0.16
* {{frac|9}} = 0.14
 
or complicated
 
* {{frac|5}}  = 0.24972497... recurring (easily rounded to 0.25)
* {{frac|7}}  = 0.186A35186A35... recurring (easily rounded to 0.187)
* {{frac|A}}  = 0.124972497... recurring (rounded to 0.125)
* {{frac|B}}  = 0.11111... recurring (rounded to 0.11)
* {{frac|11}} = 0.0B0B... recurring (rounded to 0.0B)
 
{|class="wikitable"
|-
| ''Examples in duodecimal''
| ''Decimal equivalent''
|-
| 1 × ({{frac|5|8}}) = 0.76
| 1 × ({{frac|5|8}}) = 0.625
|-
| 100 × ({{frac|5|8}}) = 76
| 144 × ({{frac|5|8}}) = 90
|-
| {{frac|576|9}} = 76
| {{frac|810|9}} = 90
|-
| {{frac|400|9}} = 54
| {{frac|576|9}} = 64
|-
| 1A.6 + 7.6 = 26
| 22.5 + 7.5 = 30
|}
|}
</center>


In 1956, Erdős proved that<ref name="Erdős1956">{{cite journal |author=[[Paul Erdős|Erdős, P.]] |year=1956 |title=On pseudoprimes and Carmichael numbers |journal=Publ. Math. Debrecen |volume=4 |pages=201–206 |url=http://www.renyi.hu/~p_erdos/1956-10.pdf |mr=79031 }}</ref>
As explained in [[recurring decimal]]s, whenever an [[irreducible fraction]] is written in [[radix point]] notation in any base, the fraction can be expressed exactly (terminates) if and only if all the [[prime factor]]s of its denominator are also prime factors of the base. Thus, in base-ten (=&nbsp;2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate:  {{frac|8}}&nbsp;=&nbsp;{{frac|(2×2×2)}}, {{frac|20}}&nbsp;=&nbsp;{{frac|(2×2×5)}} and {{frac|500}}&nbsp;=&nbsp;{{frac|(2×2×5×5×5)}} can be expressed exactly as 0.125, 0.05 and 0.002 respectively. {{frac|3}} and {{frac|7}}, however, recur (0.333... and 0.142857142857...). In the duodecimal (=&nbsp;2×2×3) system, {{frac|8}} is exact; {{frac|20}} and {{frac|500}} recur because they include 5 as a factor; {{frac|3}} is exact; and {{frac|7}} recurs, just as it does in decimal.


:<math>C(X) < X \exp\left(\frac{-k \log X \log \log \log X}{\log \log X}\right)</math>
=== Recurring digits ===


for some constant <math>k</math>. He further gave a [[heuristic argument]] suggesting that this upper bound should be close to the true growth rate of <math>C(X)</math>. The table below gives approximate minimal values for the constant ''k'' in the Erdős bound for <math>X=10^n</math> as ''n'' grows:
Arguably, factors of 3 are more commonly encountered in real-life [[division (mathematics)|division]] problems than factors of 5 (or would be, were it not for the decimal system having influenced most cultures). Thus, in practical applications, the nuisance of [[recurring decimals]] is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.


<center>
However, when recurring fractions <i>do</i> occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because [[12 (number)|12]] (twelve) is between two [[prime number]]s, [[11 (number)|11]] (eleven) and [[13 (number)|13]] (thirteen), whereas ten is adjacent to [[composite number]] [[9 (number)|9]]. Nonetheless, having a shorter or longer period doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so [[rounding]], which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor [[3 (number)|3]] in its factorization, while only one out of every five contains the prime factor [[5 (number)|5]]. All other prime factors, except 2, are not shared by either ten or twelve, so they do not
{| class="wikitable"
influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor [[2 (number)|2]] appears twice in the factorization of twelve, while only once in the factorization of ten; which means that most fractions whose denominators are [[power of two|powers of two]] will have a shorter, more convenient terminating representation in dozenal than in decimal (e.g., 1/(2<sup>2</sup>) = 0.25 <SMALL>dec</SMALL> = 0.3 <SMALL>doz</SMALL>; 1/(2<sup>3</sup>) = 0.125 <SMALL>dec</SMALL> = 0.16 <SMALL>doz</SMALL>; 1/(2<sup>4</sup>) = 0.0625 <SMALL>dec</SMALL> = 0.09 <SMALL>doz</SMALL>; 1/(2<sup>5</sup>) = 0.03125 <SMALL>dec</SMALL> = 0.046 <SMALL>doz</SMALL>; etc.).
 
{|class="wikitable"
| colspan="3" align="center" | Decimal base<br><SMALL>Prime factors of the base: <font style="color:Green">'''2'''</font>, <font style="color:Green">'''5'''</font></SMALL>
| colspan="3" align="center" | '''Duodecimal / Dozenal base'''<br><SMALL>Prime factors of the base: <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font></SMALL>
|-
| align="center" | Fraction
| align="center" | <SMALL>Prime factors<br>of the denominator<SMALL>
| align="center" | Positional representation
| align="center" | Positional representation
| align="center" | <SMALL>Prime factors<br>of the denominator<SMALL>
| align="center" | Fraction
|-
| align="center" | 1/2
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.5'''
| '''0.6'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/2
|-
| align="center" | 1/3
| align="center" | <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.'''3333... = '''0.'''{{overline|3}}
| '''0.4'''
| align="center" | <font style="color:Green">'''3'''</font>
| align="center" | 1/3
|-
| align="center" | 1/4
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.25'''
| '''0.3'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/4
|-
| align="center" | 1/5
| align="center" | <font style="color:Green">'''5'''</font>
| '''0.2'''
| bgcolor=#c0c0c0 | '''0.'''24972497... = '''0.'''{{overline|2497}}
| align="center" | <font style="color:Red">'''5'''</font>
| align="center" | 1/5
|-
| align="center" | 1/6
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.1'''{{overline|6}}
| '''0.2'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/6
|-
| align="center" | 1/7
| align="center" | <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|142857}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|186A35}}
| align="center" | <font style="color:Red">'''7'''</font>
| align="center" | 1/7
|-
| align="center" | 1/8
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.125'''
| '''0.16'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/8
|-
| align="center" | 1/9
| align="center" | <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| '''0.14'''
| align="center" | <font style="color:Green">'''3'''</font>
| align="center" | 1/9
|-
| align="center" | 1/10
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''5'''</font>
| '''0.1'''
| bgcolor=#c0c0c0 | '''0.1'''{{overline|2497}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''5'''</font>
| align="center" | 1/A
|-
| align="center" | 1/11
| align="center" | <font style="color:Red">'''11'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|09}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| align="center" | <font style="color:Red">'''B'''</font>
| align="center" | 1/B
|-
| align="center" | 1/12
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.08'''{{overline|3}}
| '''0.1'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/10
|-
| align="center" | 1/13
| align="center" | <font style="color:Red">'''13'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|076923}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0B}}
| align="center" | <font style="color:Red">'''11'''</font>
| align="center" | 1/11
|-
| align="center" | 1/14
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|714285}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|A35186}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''7'''</font>
| align="center" | 1/12
|-
| align="center" | 1/15
| align="center" | <font style="color:Red">'''3'''</font>, <font style="color:Green">'''5'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|9724}}
| align="center" | <font style="color:Green">'''3'''</font>, <font style="color:Red">'''5'''</font>
| align="center" | 1/13
|-
| align="center" | 1/16
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.0625'''
| '''0.09'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/14
|-
| align="center" | 1/17
| align="center" | <font style="color:Red">'''17'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0588235294117647}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|08579214B36429A7}}
| align="center" | <font style="color:Red">'''15'''</font>
| align="center" | 1/15
|-
| align="center" | 1/18
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|5}}
| '''0.08'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/16
|-
|-
! <math>n</math>
| align="center" | 1/19
| 4
| align="center" | <font style="color:Red">'''19'''</font>
| 6
| bgcolor=#c0c0c0 | '''0.'''{{overline|052631578947368421}}
| 8
| bgcolor=#c0c0c0 | '''0.'''{{overline|076B45}}
| 10
| align="center" | <font style="color:Red">'''17'''</font>
| 12
| align="center" | 1/17
| 14
|-
| 16
| align="center" | 1/20
| 18
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''5'''</font>
| 20
| '''0.05'''
| 21
| bgcolor=#c0c0c0 | '''0.0'''{{overline|7249}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''5'''</font>
| align="center" | 1/18
|-
| align="center" | 1/21
| align="center" | <font style="color:Red">'''3'''</font>, <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|047619}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6A3518}}
| align="center" | <font style="color:Green">'''3'''</font>, <font style="color:Red">'''7'''</font>
| align="center" | 1/19
|-
| align="center" | 1/22
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''11'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|45}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''B'''</font>
| align="center" | 1/1A
|-
| align="center" | 1/23
| align="center" | <font style="color:Red">'''23'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0434782608695652173913}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|06316948421}}
| align="center" | <font style="color:Red">'''1B'''</font>
| align="center" | 1/1B
|-
| align="center" | 1/24
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.041'''{{overline|6}}
| '''0.06'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/20
|-
| align="center" | 1/25
| align="center" | <font style="color:Green">'''5'''</font>
| '''0.04'''
| bgcolor=#c0c0c0 | '''0.'''{{overline|05915343A0B62A68781B}}
| align="center" | <font style="color:Red">'''5'''</font>
| align="center" | 1/21
|-
| align="center" | 1/26
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''13'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|384615}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|56}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''11'''</font>
| align="center" | 1/22
|-
| align="center" | 1/27
| align="center" | <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|037}}
| '''0.054'''
| align="center" | <font style="color:Green">'''3'''</font>
| align="center" | 1/23
|-
| align="center" | 1/28
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.03'''{{overline|571428}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|5186A3}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''7'''</font>
| align="center" | 1/24
|-
| align="center" | 1/29
| align="center" | <font style="color:Red">'''29'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0344827586206896551724137931}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|04B7}}
| align="center" | <font style="color:Red">'''25'''</font>
| align="center" | 1/25
|-
| align="center" | 1/30
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>, <font style="color:Green">'''5'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|3}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|4972}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>, <font style="color:Red">'''5'''</font>
| align="center" | 1/26
|-
| align="center" | 1/31
| align="center" | <font style="color:Red">'''31'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|032258064516129}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0478AA093598166B74311B28623A55}}
| align="center" | <font style="color:Red">'''27'''</font>
| align="center" | 1/27
|-
| align="center" | 1/32
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.03125'''
| '''0.046'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/28
|-
| align="center" | 1/33
| align="center" | <font style="color:Red">'''3'''</font>, <font style="color:Red">'''11'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|03}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|4}}
| align="center" | <font style="color:Green">'''3'''</font>, <font style="color:Red">'''B'''</font>
| align="center" | 1/29
|-
| align="center" | 1/34
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''17'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|2941176470588235}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|429A708579214B36}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''15'''</font>
| align="center" | 1/2A
|-
| align="center" | 1/35
| align="center" | <font style="color:Green">'''5'''</font>, <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|285714}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0414559B3931}}
| align="center" | <font style="color:Red">'''5'''</font>, <font style="color:Red">'''7'''</font>
| align="center" | 1/2B
|-
| align="center" | 1/36
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.02'''{{overline|7}}
| '''0.04'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/30
|}
 
=== Irrational numbers ===
 
As for [[irrational number]]s, none of them has a finite representation in ''any'' of the [[rational number|rational]]-based positional number systems (such as the decimal and duodecimal ones); this is because a rational-based positional number system is essentially nothing but a way of expressing quantities as a sum of fractions whose denominators are powers of the base, and by definition no ''finite'' sum of rational numbers can ever result in an irrational number. For example, 123.456 = 1 × 1/10<sup>−2</sup> + 2 × 1/10<sup>−1</sup> + 3 × 1/10<sup>0</sup> + 4 × 1/10<sup>1</sup> + 5 × 1/10<sup>2</sup> + 6 × 1/10<sup>3</sup> (this is also the reason why fractions that contain prime factors in their denominator not in common with those of the base do not have a terminating representation in that base). Moreover, the infinite series of digits of an irrational number does not exhibit a pattern of repetition; instead, the different digits succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important [[algebraic number|algebraic]] and [[transcendental number|transcendental]] irrational numbers. Some of these numbers may be perceived as having fortuitous patterns, making them easier to memorize, when represented in one base or the other.
 
{|class="wikitable"
| align="center" | ''Algebraic irrational number''
| align="center" | In decimal
| align="center" | '''In duodecimal / dozenal'''
|-
| align="center" | [[Square root of 2|√2]] <SMALL>(the length of the [[diagonal]] of a unit [[Square (geometry)|square]])</SMALL>
| 1.41421356237309... (≈ 1.4142)
| 1.4B79170A07B857... (≈ 1.5)
|-
| align="center" | [[Square root of 3|√3]] <SMALL>(the length of the diagonal of a unit [[cube]], or twice the [[height]] of an [[equilateral triangle]] of unit side)</SMALL>
| 1.73205080756887... (≈ 1.732)
| 1.894B97BB968704... (≈ 1.895)
|-
| align="center" | [[Square root of 5|√5]] <SMALL>(the length of the [[diagonal]] of a 1×2 [[rectangle]])</SMALL>
| 2.2360679774997... (≈ 2.236)
| 2.29BB132540589... (≈ 2.2A)
|-
| align="center" | [[Golden ratio|φ]] <SMALL>(phi, the golden ratio = <math>\scriptstyle \frac{1+\sqrt{5}}{2}</math>)</SMALL>
| 1.6180339887498... (≈ 1.618)
| 1.74BB6772802A4... (≈ 1.75)
|-
| align="center" | ''Transcendental irrational number''
| align="center" | In decimal
| align="center" | '''In duodecimal / dozenal'''
|-
| align="center" | ''[[Pi|π]]'' <SMALL>(pi, the ratio of [[circumference]] to [[diameter]])<SMALL>
| 3.1415926535897932384626433<br/>8327950288419716939937510...<br/>(≈ 3.1416)
| 3.184809493B918664573A6211B<br/>B151551A05729290A7809A492...<br/>(≈ 3.1848)
|-
| align="center" | [[E (mathematical constant)|e]] <SMALL>(the base of the [[natural logarithm]])</SMALL>
| 2.718281828459045... (≈ 2.718)
| 2.8752360698219B8... (≈ 2.875)
|}
 
The first few digits of the decimal and dozenal representation of another important number, the [[Euler-Mascheroni constant]] (the status of which as a rational or irrational number is not yet known), are:
 
{|class="wikitable"
| align="center" | ''Number''
| align="center" | In decimal
| align="center" | '''In duodecimal / dozenal'''
|-
|-
! ''k''
| align="center" | [[Euler-Mascheroni constant|γ]] <SMALL>(the limiting difference between the [[harmonic series (mathematics)|harmonic series]] and the natural logarithm)</SMALL>
| 2.19547
| 0.57721566490153... (~ 0.577)
| 1.97946
| 0.6B15188A6760B3... (~ 0.7)
| 1.90495
| 1.86870
| 1.86377
| 1.86293
| 1.86406
| 1.86522
| 1.86598
| 1.86619
|}
|}
</center>


In the other direction, [[W. R. (Red) Alford|Alford]], [[Andrew Granville|Granville]] and [[Carl Pomerance|Pomerance]] proved in 1994<ref name="Alford1994"/> that for sufficiently large ''X'',
==Advocacy and "dozenalism"==
:<math>C(X) > X^{2/7}.</math>
The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book ''New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics''. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized ''either'' by the adoption of ten-based weights and measure ''or'' by the adoption of the duodecimal number system.
In 2005, this bound was further improved by [[Glyn Harman|Harman]]<ref>{{cite journal |author=Glyn Harman |title=On the number of Carmichael numbers up to ''x'' |journal=Bull. Lond. Math. Soc. |volume=37 |year=2005 |pages=641–650 |doi=10.1112/S0024609305004686}}</ref> to
:<math>C(X) > X^{0.332}</math>
and then has subsequently improved the exponent to just over <math>1/3</math>.


Regarding the asymptotic distribution of Carmichael numbers, there have been several conjectures. In 1956, Erdős<ref name="Erdős1956"/> conjectured that there were <math>X^{1-o(1)}</math> Carmichael numbers for ''X'' sufficiently large. In 1981, Pomerance<ref name="Pomerance1981">{{cite journal |author=[[Carl Pomerance|Pomerance, C.]] |year=1981 |title=On the distribution of pseudoprimes |journal=Math. Comp. |volume=37 |pages=587–593|jstor=2007448}}</ref> sharpened Erdős' heuristic arguments to conjecture that there are
Rather than the symbols "A" for ten and "B" for eleven as used in [[hexadecimal]] notation and [[vigesimal]] notation (or "T" and "E" for ten and eleven), he suggested in his book and used a script X and a script E, <math>x\!</math> ([[Unicode|U+]]1D4B3) and [[Image:Scripte.png]] (U+2130), to represent the digits ten and eleven respectively, because, at least on a page of Roman script, these characters were distinct from any existing letters or numerals, yet were readily available in printers' fonts. He chose <math>x\!</math> for its resemblance to the Roman numeral X, and [[Image:Scripte.png]] as the first letter of the word "eleven".


:<math>X^{1-{\frac{\{1+o(1)\}\log\log\log X}{\log\log X}}}</math>
Another popular notation, introduced by Sir [[Isaac Pitman]], is to use a rotated 2 (resembling a script ''τ'' for "ten") to represent ten and a rotated or horizontally flipped 3 (which again resembles ''ε'') to represent eleven. This is the convention commonly employed by the Dozenal Society of Great Britain and has the advantage of being easily recognizable as digits because of their resemblance in shape to existing digits. On the other hand, the Dozenal Society of America adopted for some years the convention of using an [[asterisk]] * for ten and a [[Number sign|hash]] # for eleven. The reason was the symbol * resembles a struck-through X while # resembles a doubly-struck-through 11, and both symbols are already present in [[telephone]] [[Rotary dial|dial]]s. However, critics pointed out these symbols do not look anything like digits. Some other systems write 10 as Φ (a combination of 1 and 0) and eleven as a cross of two lines (+, x, or † for example)<!-- "+" is similar to the Chinese character for ten & "X" is the Roman numeral for ten. -->.  Problems with these symbols are evident, most notably that most of them can not be represented in the [[seven-segment display]] of most [[calculator]] displays ([[Image:Scripte.png]] being an exception, although "E" is used on calculators to indicate an [[error message]]). However, 10 and 11 do fit, both within a single digit (11 fits as is, while the 10 has to be tilted sideways, resulting in a character that resembles an O with a [[macron]], ō or <u>o</u>). A and B also fit (although B must be represented as lowercase "b" and as such, 6 must have a bar over it to distinguish the two figures) and are used on calculators for bases higher than ten.


Carmichael numbers up to ''X''. However, inside current computational ranges (such as the counts of Carmichael numbers performed by Pinch<ref name="Pinch2007"/> up to 10<sup>21</sup>), these conjectures are not yet borne out by the data.
In "Little Twelvetoes", American television series ''[[Schoolhouse Rock!]]'' portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols. ("Dek" is from the prefix "deca", "el" being short for "eleven" and "doh" an apparent shortening of "dozen".)<ref>[http://www.schoolhouserock.tv/Little.html "Little Twelvetoes"]</ref>


==Generalizations==
The Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the base-twelve system. They use the word '''dozenal''' instead of "duodecimal" because the latter comes from Latin roots that express twelve in base-ten terminology.
The notion of Carmichael number generalizes to a Carmichael ideal in any number field ''K''. For any nonzero prime ideal <math>\mathfrak p</math> in <math>{\mathcal O}_K</math>, we have <math>\alpha^{{\rm N}(\mathfrak p)} \equiv \alpha \bmod {\mathfrak p}</math> for all <math>\alpha</math> in <math>{\mathcal O}_K</math>, where <math>{\rm N}(\mathfrak p)</math> is the norm of the ideal <math>\mathfrak p</math>. (This generalizes Fermat's little theorem, that <math>m^p \equiv m \bmod p</math> for all integers ''m'' when ''p'' is prime.) Call a nonzero ideal <math>\mathfrak a</math> in <math>{\mathcal O}_K</math> Carmichael if it is not a prime ideal and <math>\alpha^{{\rm N}(\mathfrak a)} \equiv \alpha \bmod {\mathfrak a}</math> for all <math>\alpha \in {\mathcal O}_K</math>, where <math>{\rm N}(\mathfrak a)</math> is the norm of the ideal <math>\mathfrak a</math>.  When ''K'' is <math>\mathbf Q</math>, the ideal <math>\mathfrak a</math> is principal, and if we let ''a'' be its positive generator then the ideal <math>\mathfrak a = (a)</math> is Carmichael exactly when ''a'' is a Carmichael number in the usual sense.


When ''K'' is larger than the rationals it is easy to write down Carmichael ideals in <math>{\mathcal O}_K</math>: for any prime number ''p'' that splits completely in ''K'', the principal ideal <math>p{\mathcal O}_K</math> is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in <math>{\mathcal O}_K</math>. For example, if ''p'' is any prime number that is 1 mod 4, the ideal (''p'') in the Gaussian integers '''Z'''[''i''] is a Carmichael ideal.
The renowned mathematician and mental calculator [[Alexander Aitken|Alexander Craig Aitken]] was an outspoken advocate of the advantages and superiority of duodecimal over decimal:
{{quote|The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten. Anyone having these tables at command will do these calculations more than one-and-a-half times as fast in the duodecimal scale as in the decimal. This is my experience; I am certain that even more so it would be the experience of others.|A. C. Aitken|in ''The Listener'', January 25, 1962<ref>[http://www.dozenalsociety.org.uk/leafletsetc/aitken.html Basic Stuff<!-- Bot generated title -->]</ref>}}


Both prime and Carmichael numbers satisfy the following equality:
{{quote|But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal.|A. C. Aitken|''The Case Against Decimalisation'' (Edinburgh / London: Oliver & Boyd, 1962)<ref>[http://www.dozenalsociety.org.uk/pdfs/aitken.pdf The Case against Decimalisation<!-- Bot generated title -->]</ref>}}
:<math>\gcd (\sum_{x=1}^{n-1} x^{n-1}, n)\equiv 1</math>


==Higher-order Carmichael numbers==
In [[Leo Frankowski]]'s [[Conrad Stargard]] novels, Conrad introduces a duodecimal system of arithmetic at the suggestion of a merchant, who is accustomed to buying and selling goods in dozens and grosses, rather than tens or hundreds. He then invents an entire system of weights and measures in base twelve, including a clock with twelve hours in a day, rather than twenty-four hours.
Carmichael numbers can be generalized using concepts of [[abstract algebra]].


The above definition states that a composite integer ''n'' is Carmichael
In [[Lee Carroll]]'s ''Kryon: Alchemy of the Human Spirit'', a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by [[Kryon]] (one of the widely popular [[New Age]] channeled entities) for all-round use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the [[golden ratio]], as well as provides numerous number symmetry-based arguments for the universal nature of the base-12 number system.<ref>''[https://www.kryon.com/k_13.html Kryon—Alchemy of the Human Spirit]'', ISBN 0-9636304-8-2</ref>
precisely when the ''n''th-power-raising function ''p''<sub>''n''</sub> from the [[ring (mathematics)|ring]] '''Z'''<sub>''n''</sub> of integers modulo ''n'' to itself is the identity function. The identity is the only '''Z'''<sub>''n''</sub>-[[algebra over a field|algebra]] [[endomorphism]] on '''Z'''<sub>''n''</sub> so we can restate the definition as asking that ''p''<sub>''n''</sub> be an algebra endomorphism of '''Z'''<sub>''n''</sub>.
As above, ''p''<sub>''n''</sub> satisfies the same property whenever ''n'' is prime.


The ''n''th-power-raising function ''p''<sub>''n''</sub> is also defined on any '''Z'''<sub>''n''</sub>-algebra '''A'''. A theorem states that ''n'' is prime if and only if all such functions ''p''<sub>''n''</sub> are algebra endomorphisms.
=== Dozenal clock ===


In-between these two conditions lies the definition of '''Carmichael number of order m''' for any positive integer ''m'' as any composite number ''n'' such that ''p''<sub>''n''</sub> is an endomorphism on every '''Z'''<sub>''n''</sub>-algebra that can be generated as '''Z'''<sub>''n''</sub>-[[module (mathematics)|module]] by ''m'' elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.
[http://www.dozenalsociety.org.uk/apps/dozenalclock.html Dozenal Clock by Joshua Harkey]


===Properties===
=== Dozenal metric systems ===
Korselt's criterion can be generalized to higher-order Carmichael numbers, as shown by Howe.<ref>Everett W. Howe. [http://arxiv.org/abs/math.NT/9812089 "Higher-order Carmichael numbers."] ''Mathematics of Computation'' '''69''' (2000), pp. 1711–1719.</ref>


A heuristic argument, given in the same paper, appears to suggest that there are infinitely many Carmichael numbers of order ''m'', for any ''m''. However, not a single Carmichael number of order 3 or above is known.
[[Systems of measurement]] proposed by dozenalists include:


==Notes==
* Tom Pendlebury's [[TGM (measurement system)|TGM]] system<ref>{{cite web|last=Pendlebury|first=Tom|title=TGM|url=http://www.dozenalsociety.org.uk/pdfs/TGMbooklet.pdf}}</ref>
* Takashi Suga's [[Universal Unit System]]<ref>{{cite web|last=Suga|first=Takashi|title=Universal Unit System|url=http://www.asahi-net.or.jp/~dd6t-sg/univunit-e/}}</ref>
 
== See also ==
 
*[[Senary]] (base 6)
*[[base 24|Quadrovigesimal]] (base 24)
*[[base 36|Hexatridecimal]] (base 36)
*[[Sexagesimal]] (base 60)
*[[Babylonian numerals]]
 
== References ==
{{reflist}}
{{reflist}}


==References==
== External links ==
*{{cite journal |author=Carmichael, R. D.|year=1910|title=Note on a new number theory function |journal=[[Bulletin of the American Mathematical Society]] |volume=16 |issue=5|pages=232–238 |url=http://www.ams.org/journals/bull/1910-16-05/home.html}}
*[http://www.dozenal.org/ Dozenal Society of America]
*{{cite journal |author=Carmichael, R. D. |year=1912 |title=On composite numbers ''P'' which satisfy the Fermat congruence <math>a^{P-1}\equiv 1\bmod P</math> |journal=[[American Mathematical Monthly]] |volume=19 |issue=2 |pages=22–27 |doi=10.2307/2972687}}
*[http://www.dozenalsociety.org.uk/ Dozenal Society of Great Britain website]
*{{cite journal |author=Chernick, J. |year=1939 |title=On Fermat's simple theorem |journal=Bull. Amer. Math. Soc. |volume=45 |pages=269–274 |doi=10.1090/S0002-9904-1939-06953-X  |url=http://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf}}
*[http://flud.org/dozenal-calc.html Online Decimal-Dozenal Converter]
*{{cite journal |author=Korselt, A. R. |year=1899 |title=Problème chinois |journal=L'intermédiaire des mathématiciens |volume=6 |pages=142–143}}
*[http://7r4n5.com/papers/bases-of-counting/ The Bases of Counting]
*{{cite journal |author=Löh, G.; Niebuhr, W. |year=1996 |url=http://www.ams.org/mcom/1996-65-214/S0025-5718-96-00692-8/S0025-5718-96-00692-8.pdf |title=A new algorithm for constructing large Carmichael numbers |journal=Math. Comp. |volume=65 |pages=823–836 |doi=10.1090/S0025-5718-96-00692-8}}
*{{cite book | title = The Book of Prime Number Records | publisher = Springer | year = 1989 | isbn = 978-0-387-97042-4 | author = [[Paulo Ribenboim|Ribenboim, P.]] }}
*{{cite journal |author=Šimerka, V.|year=1885 |title=Zbytky z arithmetické posloupnosti (On the remainders of an arithmetic progression) |journal=Časopis pro pěstování matematiky a fysiky |volume=14 |issue=5 |pages=221–225 |url=http://dml.cz/handle/10338.dmlcz/122245}}
 
==External links==
*[http://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Carmichael-Zahlen Table of Carmichael numbers]
*[http://www.kobepharma-u.ac.jp/~math/notes/note02.html Carmichael numbers up to 10^12]
*{{MathPages|id=home/kmath028/kmath028|title=The Dullness of 1729}}
*{{MathWorld | urlname=CarmichaelNumber | title=Carmichael Number}}
*[http://www.numericana.com/answer/modular.htm Final Answers Modular Arithmetic]


[[Category:Integer sequences]]
[[Category:Positional numeral systems]]
[[Category:Modular arithmetic]]
[[Category:Pseudoprimes]]


[[ar:عدد كارميكائيل]]
[[ar:نظام عد ثنائي عشر]]
[[ca:Nombres de Carmichael]]
[[ca:Sistema duodecimal]]
[[cs:Carmichaelovo číslo]]
[[da:Duodecimal]]
[[de:Carmichael-Zahl]]
[[de:Duodezimalsystem]]
[[es:Número de Carmichael]]
[[es:Sistema duodecimal]]
[[eo:Nombro de Carmichael]]
[[eo:Dekduuma sistemo]]
[[fr:Nombre de Carmichael]]
[[eu:Zenbaki-sistema hamabitar]]
[[ko:카마이클 수]]
[[fa:دستگاه اعداد پایه ۱۲]]
[[it:Numero di Carmichael]]
[[fr:Système duodécimal]]
[[he:מספר קרמייקל]]
[[ko:십이진법]]
[[nl:Carmichael-getal]]
[[is:Tylftakerfi]]
[[ja:カーマイケル数]]
[[he:בסיס דואודצימלי]]
[[pl:Liczby Carmichaela]]
[[hu:Tizenkettes számrendszer]]
[[pt:Número de Carmichael]]
[[nl:Twaalftallig stelsel]]
[[ru:Число Кармайкла]]
[[ja:十二進法]]
[[simple:Carmichael number]]
[[no:Tolvtallsystemet]]
[[sl:Carmichaelovo število]]
[[nn:Tolvtalsystemet]]
[[fi:Carmichaelin luku]]
[[pl:Dwunastkowy system liczbowy]]
[[sv:Carmichaeltal]]
[[pt:Sistema de numeração duodecimal]]
[[uk:Число Кармайкла]]
[[ru:Двенадцатеричная система счисления]]
[[zh:卡邁克爾數]]
[[sl:Dvanajstiški številski sistem]]
[[fi:Duodesimaalijärjestelmä]]
[[sv:Duodecimalsystemet]]
[[th:เลขฐานสิบสอง]]
[[tr:On ikili sayı sistemi]]
[[uk:Дванадцяткова система числення]]
[[zh:十二进制]]

Revision as of 11:15, 8 August 2014

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HostGator has actually proven it is one of the leaders in the internet hosting market. HostGator provides top of the line customer service and dependability. In case you loved this post and you would like to receive details with regards to Hostgator Coupons kindly visit our web-site. It is no error that HostGator is one of the fastest expanding companies in America. The duodecimal system (also known as base-12 or dozenal) is a positional notation numeral system using twelve as its base. In this system, the number ten may be written as "A", "T" or "X", and the number eleven as "B" or "E" (another common notation, introduced by Sir Isaac Pitman, is to use a rotated "2" for ten and a reversed "3" for eleven). The number twelve (that is, the number written as "12" in the base ten numerical system) is instead written as "10" in duodecimal (meaning "1 dozen and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (i.e. the same number that in decimal is written as "14"). Similarly, in duodecimal "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").

The number twelve, a highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal is sometimes named the number system with the most optimal radix economy.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9...) have a terminating representation in duodecimal. In particular, the five most elementary fractions (½, ⅓, ⅔, ¼ and ¾) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (since it is the least common multiple of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems, although the sexagesimal system (where the reciprocals of all 5-smooth numbers terminate) does better in this respect (but at the cost of an unwieldy multiplication table).

Origin

In this section, numerals are based on decimal places. For example, 10 means ten, 12 means twelve.

Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara;[1] the Chepang_language of Nepal[2] and the Mahl language of Minicoy Island in India are known to use duodecimal numerals. In fiction, J. R. R. Tolkien's Elvish languages use a hybrid decimal-duodecimal system, primarily decimal but with special names for multiples of six.

Germanic languages have special words for 11 and 12, such as eleven and twelve in English, which are often misinterpreted as vestiges of a duodecimal system.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.

Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and the Babylonians had twelve hours in a day (although at some point this was changed to 24). Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches.

Being a versatile denominator in fractions may explain why we have 12 inches in an imperial foot, 12 ounces in a troy pound, 12 old British pence in a shilling, 24 (12×2) hours in a day, and many other items counted by the dozen, gross (144, square of 12) or great gross (1728, cube of 12). The Romans used a fraction system based on 12, including the uncia which became both the English words ounce and inch. Pre-decimalisation, the United Kingdom and Republic of Ireland used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling or Irish pound), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.

The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers).[3] It is possible to count to 12 with your thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.[4][5]

Places

In a duodecimal place system, ten can be written as A, eleven can be written as B, and twelve is written as 10. For alternative symbols, see below.

According to this notation, duodecimal 50 expresses the same quantity as decimal 60 (= five times twelve), duodecimal 60 is equivalent to decimal 72 (= six times twelve = half a gross), duodecimal 100 has the same value as decimal 144 (= twelve times twelve = one gross), etc.

Comparison to other numeral systems

A duodecimal multiplication table
1 2 3 4 5 6 7 8 9 A B 10
2 4 6 8 A 10 12 14 16 18 1A 20
3 6 9 10 13 16 19 20 23 26 29 30
4 8 10 14 18 20 24 28 30 34 38 40
5 A 13 18 21 26 2B 34 39 42 47 50
6 10 16 20 26 30 36 40 46 50 56 60
7 12 19 24 2B 36 41 48 53 5A 65 70
8 14 20 28 34 40 48 54 60 68 74 80
9 16 23 30 39 46 53 60 69 76 83 90
A 18 26 34 42 50 5A 68 76 84 92 A0
B 1A 29 38 47 56 65 74 83 92 A1 B0
10 20 30 40 50 60 70 80 90 A0 B0 100

The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are prime. The decimal system has only four factors, which are 1, 2, 5, and 10; of which 2 and 5 are prime. Vigesimal adds two factors to those of ten, namely 4 and 20, but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base (i.e., the digit set and the multiplication table are much larger). Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal has five factors, adding 4, 8 and 16 to those of 2, but no additional prime. Trigesimal is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30), the smallest system that has four different prime factors is Base 210 and the pattern follows the primorials. Sexagesimal—which the ancient Sumerians and Babylonians among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors.Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

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Conversion tables to and from decimal

To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any dozenal number between 0.01 and BBB,BBB.BB to decimal, or any decimal number between 0.01 and 999,999.99 to dozenal. To use them, we first decompose the given number into a sum of numbers with only one significant digit each. For example:

123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08

This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then we use the digit conversion tables to obtain the equivalent value in the target base for each digit. If the given number is in dozenal and the target base is decimal, we get:

(dozenal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58Template:Overline333333333... + 0.0Template:Overline5555555555...

Now, since the summands are already converted to base ten, we use the usual decimal arithmetic to perform the addition and recompose the number, arriving at the conversion result: 

  Dozenal  ----->  Decimal
  
  100,000     =    248,832
   20,000     =     41,472
    3,000     =      5,184
      400     =        576
       50     =         60
 +      6     =   +      6
        0.7   =          0.58Template:Overline333333333...
        0.08  =          0.0Template:Overline5555555555...
--------------------------------------------
  123,456.78  =    296,130.63Template:Overline888888888...

That is, (dozenal) 123,456.78 equals (decimal) 296,130.63Template:Overline888888888... ≈ 296,130.64

If the given number is in decimal and the target base is dozenal, the method is basically same. Using the digit conversion tables:

(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (dozenal) 49,A54 + B,6A8 + 1,8A0 + 294 + 42 + 6 + 0.8Template:Overline4972497249724972497... + 0.Template:Overline0B62...

However, in order to do this sum and recompose the number, we now have to use the addition tables for dozenal, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in dozenal as well. In decimal, 6 + 6 equals 12, but in dozenal it equals 10; so if we used decimal arithmetic with dozenal numbers we would arrive at an incorrect result. Doing the arithmetic properly in dozenal, we get the result: 

  Decimal  ----->  Dozenal
  
  100,000     =     49,A54
   20,000     =      B,6A8
    3,000     =      1,8A0
      400     =        294
       50     =         42
 +      6     =   +      6
        0.7   =          0.8Template:Overline4972497249724972497...
        0.08  =          0.Template:Overline0B62...
--------------------------------------------------------
  123,456.78  =     5B,540.9Template:Overline43A...

That is, (decimal) 123,456.78 equals (dozenal) 5B,540.9Template:Overline43A... ≈ 5B,540.94

Dozenal to decimal digit conversion

Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec.
100,000 248,832 10,000 20,736 1,000 1,728 100 144 10 12 1 1 0.1 0.08Template:Overline 0.01 0.0069Template:Overline
200,000 497,664 20,000 41,472 2,000 3,456 200 288 20 24 2 2 0.2 0.1Template:Overline 0.02 0.013Template:Overline
300,000 746,496 30,000 62,208 3,000 5,184 300 432 30 36 3 3 0.3 0.25 0.03 0.0208Template:Overline
400,000 995,328 40,000 82,944 4,000 6,912 400 576 40 48 4 4 0.4 0.Template:Overline 0.04 0.02Template:Overline
500,000 1,244,160 50,000 103,680 5,000 8,640 500 720 50 60 5 5 0.5 0.41Template:Overline 0.05 0.0347Template:Overline
600,000 1,492,992 60,000 124,416 6,000 10,368 600 864 60 72 6 6 0.6 0.5 0.06 0.041Template:Overline
700,000 1,741,824 70,000 145,152 7,000 12,096 700 1008 70 84 7 7 0.7 0.58Template:Overline 0.07 0.0486Template:Overline
800,000 1,990,656 80,000 165,888 8,000 13,824 800 1152 80 96 8 8 0.8 0.Template:Overline 0.08 0.0Template:Overline
900,000 2,239,488 90,000 186,624 9,000 15,552 900 1,296 90 108 9 9 0.9 0.75 0.09 0.0625
A00,000 2,488,320 A0,000 207,360 A,000 17,280 A00 1,440 A0 120 A 10 0.A 0.8Template:Overline 0.0A 0.069Template:Overline
B00,000 2,737,152 B0,000 228,096 B,000 19,008 B00 1,584 B0 132 B 11 0.B 0.91Template:Overline 0.0B 0.0763Template:Overline

Decimal to dozenal digit conversion

Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz.
100,000 49,A54 10,000 5,954 1,000 6B4 100 84 10 A 1 1 0.1 0.1Template:Overline 0.01 0.0Template:Overline
200,000 97,8A8 20,000 B,6A8 2,000 1,1A8 200 148 20 18 2 2 0.2 0.Template:Overline 0.02 0.0Template:Overline
300,000 125,740 30,000 15,440 3,000 1,8A0 300 210 30 26 3 3 0.3 0.3Template:Overline 0.03 0.0Template:Overline
400,000 173,594 40,000 1B,194 4,000 2,394 400 294 40 34 4 4 0.4 0.Template:Overline 0.04 0.0Template:Overline
500,000 201,428 50,000 24,B28 5,000 2,A88 500 358 50 42 5 5 0.5 0.6 0.05 0.0Template:Overline
600,000 24B,280 60,000 2A,880 6,000 3,580 600 420 60 50 6 6 0.6 0.Template:Overline 0.06 0.0Template:Overline
700,000 299,114 70,000 34,614 7,000 4,074 700 4A4 70 5A 7 7 0.7 0.8Template:Overline 0.07 0.0Template:Overline
800,000 326,B68 80,000 3A,368 8,000 4,768 800 568 80 68 8 8 0.8 0.Template:Overline 0.08 0.Template:Overline
900,000 374,A00 90,000 44,100 9,000 5,260 900 630 90 76 9 9 0.9 0.ATemplate:Overline 0.09 0.1Template:Overline

Conversion of powers

Exponent Powers of 2 Powers of 3 Powers of 4 Powers of 5 Powers of 6 Powers of 7
Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz.
^6 64 54 729 509 4,096 2454 15,625 9,061 46,656 23,000 117,649 58,101
^5 32 28 243 183 1,024 714 3,125 1,985 7,776 4,600 16,807 9,887
^4 16 14 81 69 256 194 625 441 1,296 900 2,401 1,481
^3 8 8 27 23 64 54 125 A5 216 160 343 247
^2 4 4 9 9 16 14 25 21 36 30 49 41
^1 2 2 3 3 4 4 5 5 6 6 7 7
^−1 0.5 0.6 0.Template:Overline 0.4 0.25 0.3 0.2 0.Template:Overline 0.1Template:Overline 0.2 0.Template:Overline 0.Template:Overline
^−2 0.25 0.3 0.Template:Overline 0.14 0.0625 0.09 0.04 0.Template:Overline 0.02Template:Overline 0.04 0.Template:Overline 0.Template:Overline
Exponent Powers of 8 Powers of 9 Powers of 10 Powers of 11 Powers of 12
Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz. Dec. Doz.
^6 262,144 107,854 531,441 217,669 1,000,000 402,854 1,771,561 715,261 2,985,984 1,000,000
^5 32,768 16,B68 59,049 2A,209 100,000 49,A54 161,051 79,24B 248,832 100,000
^4 4,096 2,454 6,561 3,969 10,000 5,954 14,641 8,581 20,736 10,000
^3 512 368 729 509 1,000 6B4 1,331 92B 1,728 1,000
^2 64 54 81 69 100 84 121 A1 144 100
^1 8 8 9 9 10 A 11 B 12 10
^−1 0.125 0.16 0.Template:Overline 0.14 0.1 0.1Template:Overline 0.Template:Overline 0.Template:Overline 0.08Template:Overline 0.1
^−2 0.015625 0.023 0.Template:Overline 0.0194 0.01 0.0Template:Overline 0.Template:Overline 0.Template:Overline 0.0069Template:Overline 0.01

Fractions and irrational numbers

Fractions

Duodecimal fractions may be simple:

or complicated

Examples in duodecimal Decimal equivalent
1 × (Template:Frac) = 0.76 1 × (Template:Frac) = 0.625
100 × (Template:Frac) = 76 144 × (Template:Frac) = 90
Template:Frac = 76 Template:Frac = 90
Template:Frac = 54 Template:Frac = 64
1A.6 + 7.6 = 26 22.5 + 7.5 = 30

As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-ten (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: Template:Frac = Template:Frac, Template:Frac = Template:Frac and Template:Frac = Template:Frac can be expressed exactly as 0.125, 0.05 and 0.002 respectively. Template:Frac and Template:Frac, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, Template:Frac is exact; Template:Frac and Template:Frac recur because they include 5 as a factor; Template:Frac is exact; and Template:Frac recurs, just as it does in decimal.

Recurring digits

Arguably, factors of 3 are more commonly encountered in real-life division problems than factors of 5 (or would be, were it not for the decimal system having influenced most cultures). Thus, in practical applications, the nuisance of recurring decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.

However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12 (twelve) is between two prime numbers, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to composite number 9. Nonetheless, having a shorter or longer period doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so rounding, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, while only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor 2 appears twice in the factorization of twelve, while only once in the factorization of ten; which means that most fractions whose denominators are powers of two will have a shorter, more convenient terminating representation in dozenal than in decimal (e.g., 1/(22) = 0.25 dec = 0.3 doz; 1/(23) = 0.125 dec = 0.16 doz; 1/(24) = 0.0625 dec = 0.09 doz; 1/(25) = 0.03125 dec = 0.046 doz; etc.).

Decimal base
Prime factors of the base: 2, 5 		Duodecimal / Dozenal base
Prime factors of the base: 2, 3
Fraction Prime factors
of the denominator 		Positional representation Positional representation Prime factors
of the denominator 		Fraction
1/2 2 0.5 0.6 2 1/2
1/3 3 0.3333... = 0.Template:Overline 0.4 3 1/3
1/4 2 0.25 0.3 2 1/4
1/5 5 0.2 0.24972497... = 0.Template:Overline 5 1/5
1/6 2, 3 0.1Template:Overline 0.2 2, 3 1/6
1/7 7 0.Template:Overline 0.Template:Overline 7 1/7
1/8 2 0.125 0.16 2 1/8
1/9 3 0.Template:Overline 0.14 3 1/9
1/10 2, 5 0.1 0.1Template:Overline 2, 5 1/A
1/11 11 0.Template:Overline 0.Template:Overline B 1/B
1/12 2, 3 0.08Template:Overline 0.1 2, 3 1/10
1/13 13 0.Template:Overline 0.Template:Overline 11 1/11
1/14 2, 7 0.0Template:Overline 0.0Template:Overline 2, 7 1/12
1/15 3, 5 0.0Template:Overline 0.0Template:Overline 3, 5 1/13
1/16 2 0.0625 0.09 2 1/14
1/17 17 0.Template:Overline 0.Template:Overline 15 1/15
1/18 2, 3 0.0Template:Overline 0.08 2, 3 1/16
1/19 19 0.Template:Overline 0.Template:Overline 17 1/17
1/20 2, 5 0.05 0.0Template:Overline 2, 5 1/18
1/21 3, 7 0.Template:Overline 0.0Template:Overline 3, 7 1/19
1/22 2, 11 0.0Template:Overline 0.0Template:Overline 2, B 1/1A
1/23 23 0.Template:Overline 0.Template:Overline 1B 1/1B
1/24 2, 3 0.041Template:Overline 0.06 2, 3 1/20
1/25 5 0.04 0.Template:Overline 5 1/21
1/26 2, 13 0.0Template:Overline 0.0Template:Overline 2, 11 1/22
1/27 3 0.Template:Overline 0.054 3 1/23
1/28 2, 7 0.03Template:Overline 0.0Template:Overline 2, 7 1/24
1/29 29 0.Template:Overline 0.Template:Overline 25 1/25
1/30 2, 3, 5 0.0Template:Overline 0.0Template:Overline 2, 3, 5 1/26
1/31 31 0.Template:Overline 0.Template:Overline 27 1/27
1/32 2 0.03125 0.046 2 1/28
1/33 3, 11 0.Template:Overline 0.0Template:Overline 3, B 1/29
1/34 2, 17 0.0Template:Overline 0.0Template:Overline 2, 15 1/2A
1/35 5, 7 0.0Template:Overline 0.Template:Overline 5, 7 1/2B
1/36 2, 3 0.02Template:Overline 0.04 2, 3 1/30

Irrational numbers

As for irrational numbers, none of them has a finite representation in any of the rational-based positional number systems (such as the decimal and duodecimal ones); this is because a rational-based positional number system is essentially nothing but a way of expressing quantities as a sum of fractions whose denominators are powers of the base, and by definition no finite sum of rational numbers can ever result in an irrational number. For example, 123.456 = 1 × 1/10−2 + 2 × 1/10−1 + 3 × 1/100 + 4 × 1/101 + 5 × 1/102 + 6 × 1/103 (this is also the reason why fractions that contain prime factors in their denominator not in common with those of the base do not have a terminating representation in that base). Moreover, the infinite series of digits of an irrational number does not exhibit a pattern of repetition; instead, the different digits succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important algebraic and transcendental irrational numbers. Some of these numbers may be perceived as having fortuitous patterns, making them easier to memorize, when represented in one base or the other.

Algebraic irrational number In decimal In duodecimal / dozenal
√2 (the length of the diagonal of a unit square) 1.41421356237309... (≈ 1.4142) 1.4B79170A07B857... (≈ 1.5)
√3 (the length of the diagonal of a unit cube, or twice the height of an equilateral triangle of unit side) 1.73205080756887... (≈ 1.732) 1.894B97BB968704... (≈ 1.895)
√5 (the length of the diagonal of a 1×2 rectangle) 2.2360679774997... (≈ 2.236) 2.29BB132540589... (≈ 2.2A)
φ (phi, the golden ratio = ) 1.6180339887498... (≈ 1.618) 1.74BB6772802A4... (≈ 1.75)
Transcendental irrational number In decimal In duodecimal / dozenal
π (pi, the ratio of circumference to diameter) 3.1415926535897932384626433
8327950288419716939937510...
(≈ 3.1416) 		3.184809493B918664573A6211B
B151551A05729290A7809A492...
(≈ 3.1848)
e (the base of the natural logarithm) 2.718281828459045... (≈ 2.718) 2.8752360698219B8... (≈ 2.875)

The first few digits of the decimal and dozenal representation of another important number, the Euler-Mascheroni constant (the status of which as a rational or irrational number is not yet known), are:

Number In decimal In duodecimal / dozenal
γ (the limiting difference between the harmonic series and the natural logarithm) 0.57721566490153... (~ 0.577) 0.6B15188A6760B3... (~ 0.7)

Advocacy and "dozenalism"

The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of ten-based weights and measure or by the adoption of the duodecimal number system.

Rather than the symbols "A" for ten and "B" for eleven as used in hexadecimal notation and vigesimal notation (or "T" and "E" for ten and eleven), he suggested in his book and used a script X and a script E, (U+1D4B3) and File:Scripte.png (U+2130), to represent the digits ten and eleven respectively, because, at least on a page of Roman script, these characters were distinct from any existing letters or numerals, yet were readily available in printers' fonts. He chose for its resemblance to the Roman numeral X, and File:Scripte.png as the first letter of the word "eleven".

Another popular notation, introduced by Sir Isaac Pitman, is to use a rotated 2 (resembling a script τ for "ten") to represent ten and a rotated or horizontally flipped 3 (which again resembles ε) to represent eleven. This is the convention commonly employed by the Dozenal Society of Great Britain and has the advantage of being easily recognizable as digits because of their resemblance in shape to existing digits. On the other hand, the Dozenal Society of America adopted for some years the convention of using an asterisk * for ten and a hash # for eleven. The reason was the symbol * resembles a struck-through X while # resembles a doubly-struck-through 11, and both symbols are already present in telephone dials. However, critics pointed out these symbols do not look anything like digits. Some other systems write 10 as Φ (a combination of 1 and 0) and eleven as a cross of two lines (+, x, or † for example). Problems with these symbols are evident, most notably that most of them can not be represented in the seven-segment display of most calculator displays (File:Scripte.png being an exception, although "E" is used on calculators to indicate an error message). However, 10 and 11 do fit, both within a single digit (11 fits as is, while the 10 has to be tilted sideways, resulting in a character that resembles an O with a macron, ō or o). A and B also fit (although B must be represented as lowercase "b" and as such, 6 must have a bar over it to distinguish the two figures) and are used on calculators for bases higher than ten.

In "Little Twelvetoes", American television series Schoolhouse Rock! portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols. ("Dek" is from the prefix "deca", "el" being short for "eleven" and "doh" an apparent shortening of "dozen".)[6]

The Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the base-twelve system. They use the word dozenal instead of "duodecimal" because the latter comes from Latin roots that express twelve in base-ten terminology.

The renowned mathematician and mental calculator Alexander Craig Aitken was an outspoken advocate of the advantages and superiority of duodecimal over decimal: 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

In Leo Frankowski's Conrad Stargard novels, Conrad introduces a duodecimal system of arithmetic at the suggestion of a merchant, who is accustomed to buying and selling goods in dozens and grosses, rather than tens or hundreds. He then invents an entire system of weights and measures in base twelve, including a clock with twelve hours in a day, rather than twenty-four hours.

In Lee Carroll's Kryon: Alchemy of the Human Spirit, a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by Kryon (one of the widely popular New Age channeled entities) for all-round use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the golden ratio, as well as provides numerous number symmetry-based arguments for the universal nature of the base-12 number system.[7]

Dozenal clock

Dozenal Clock by Joshua Harkey

Dozenal metric systems

Systems of measurement proposed by dozenalists include:

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

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  1. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  2. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  3. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  4. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.
  5. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  6. ↑ "Little Twelvetoes"
  7. ↑ Kryon—Alchemy of the Human Spirit, ISBN 0-9636304-8-2
  8. ↑ Template:Cite web
  9. ↑ Template:Cite web
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